# Equivalence of semidefinite decomposition?

If we have an $$n \times n$$ positive semidefinite matrix $$A$$ and we have two decompositions such that $$A = B B^T = C C^T$$ for some $$n \times n$$ matrices $$B$$ and $$C$$.

Is it true that $$B$$ and $$C$$ are related by a unitary matrix? Specifically, is $$B U = C$$ for some unitary matrix $$U$$?

• I guess no. Try to find a counterexample by choosing $A=\pmatrix{1&0\\0&0}$. – Berci Jan 23 at 9:06